Braking index of isolated pulsars: open questions and ways forward
aa r X i v : . [ a s t r o - ph . H E ] M a r Compact Stars in the QCD Phase Diagram IV (CSQCD IV)September 26-30, 2014, Prerow, Germany
Braking index of isolated pulsars: open questionsand ways forward
Oliver Hamil Department of Physics and Astronomy, University of Tennessee, Knoxville,Tennessee 37996 USA
Isolated pulsars are rotating neutron stars with accurately measured angular velocitiesΩ, and their time derivatives which show unambiguously that the pulsars are slowingdown. Although the exact mechanism of the spin-down is a question of debate indetail, the commonly accepted view is that it arises through emission of magneticdipole radiation (MDR) from a rotating magnetized body. Other processes, includingthe emission of gravitational radiation, and of relativistic particles (pulsar wind), arealso being considered. The calculated energy loss by a rotating pulsar with a constantmoment of inertia is assumed proportional to a model dependent power of Ω. Thisrelation leads to the power law ˙Ω = -K Ω n where n is called the braking index. TheMDR model predicts n exactly equal to 3. Selected observations of isolated pulsarsprovide rather precise values of n , individually accurate to a few percent or better,in the range 1 < n < The pulsar spin-down rate is an observation that has been recorded for the past fewdecades. That pulsars can be seen and measured is evidence of the electromagneticradiation emanating from what is assumed to be a strong dipole misaligned with theaxis of rotation. This misaligned dipole is generally considered to be the source of thepulsar spin-down as it accounts for rotational energy being carried away from the star[1, 2, 3, 4, 5]. The observed loss of energy can be modeled by the simple power law,˙Ω = -K Ω n where n is the so-called the braking index. The braking index itself is apurely observational parameter which is determined from pulsar timing observationsalong with their first and second derivatives [6] n = Ω ¨Ω˙Ω . (1)1he value for the braking index n is theoretically determined by the torque mech-anism working against the rotation of the star. In the simple MDR model, theradiating dipole carries energy away from rotation thereby producing a torque whichslows the rate of rotation of the pulsar. This is the main assumption for pulsarsbecause of the nature of the observed radiation, but there are at least two other ac-ceptable possibilities for the torque mechanism. The emission of charged particles,accelerated to relativistic velocities, forming a massive wind from the surface of thepulsar [7, 8] may play a role in spin-down. Another consideration is higher ordermultipole electromagnetic radiation, or gravitational quadropole components to theradiated energy [9, 10]. For each mechanism described above, the theoretical brakingindex is n = 1 , , Of the mechanisms described above, the most readily accepted is the MDR model.This model predicts the radiated energy expected from a magnetized sphere rotatingin vacuum (the accepted description of pulsars). The braking index value for theMDR model is n = 3. We see in Table 1 eight pulsars with accurately observed spinevolution. The given values for the braking index are accurate to within a few percentor better. It is clear that none of these observed stars has a braking index consistentwith any of the values produced from theory.PSR Frequency n Ref.(Hz)B1509 −
58 6.633598804 2.839 ± − ± − ± ± ± −
69 19.8344965 2.140 ± − ± −
45 (Vela) 11.2 1.4 ± − ± E = − C Ω n +1 , (2)2here C contains the physics of the associated mechanism, Ω is the rotational velocity,and n is the braking index. Likewise, the rotational energy of a rotating sphere isgiven by, ˙ E = ddt ( 12 I Ω ) , (3)where I is the moment of inertia. Setting the above two equations equal to each otherleads to two different treatments of the result. Firstly, we can assume the moment ofinertia is constant in time which leads to the known power law,˙Ω = − K Ω n , (4)where K = C/I . This leads to the given values described earlier of n = 1 , ,
5. Asecond approach is to consider that the moment of inertia changes in time which leadsto braking index as a function of Ω, n (Ω) = n − I ′ + Ω I ′′ I + Ω I ′ + C ′ Ω C , (5)where n is the theoretical braking index associated with each torque mechanism, andthe prime notation denotes derivatives with respect to angular frequency.As we see from the above equations, the simple values for braking index comefrom assumptions that the moment of inertia of the star is constant (or that thestar is static), and the parameters defining the physics of the braking torque are allconstant. This means that composition, magnetic field, rotational effects, etc. haveno effect on braking. The frequency dependent solution (Eq. 5) allows for the problemto be explored in much greater detail. The microscopic proterties of the star must beconsidered in order to have a cohesive model which allows for changing values whichmay affect the braking index. The frequency dependent solution to the braking index allows for the problem to beapproached from two extreme positions. There can be a dependence of frequencywhere the moment of intertia changes at a rate which brings the braking index awayfrom the canonical value, or in the limit of very slow rotation, the parameters govern-ing the torque mechanism can be changing with frequency (or time). Close examina-tion of Eq. 5 shows that changing moment of inertia can have a significant effect onthe braking index at high frequencies, and in the limit that moment of inertia changesvery slowly with frequency, there is still a term allowing for the physics in the torquemechanism ( C ) to change. In the assumed MDR model, high frequency calculationsshow a reduction in braking index consistent with observation; however, the observed3ulsars are all rotating at low frequency. It is shown in Figure 1 the effect of chang-ing moment of inertia over a range of frequency up to 160 Hz assuming constant C .From the figure it is clear that below this frequency, the change in moment of inertiais negligible, and the subsequent deviation from n = 3 is also insignificant. Notingthat the data in Table 1 spans a range of frequencies between about 1 - 30 Hz, it isworthwhile to consider changes in the physical parameters of the torque mechanismsat low frequency. Frequency (Hz) B r a k i ng I nd e x o r R p / R e q p /R eq )Lower limit Figure 1: Braking index as a function of frequency (solid line) compared with the ratiobetween polar (R p ) and equatorial radii (R eq ), normalized to three, which determinesdeformation of the star. The difference between the two lines represents a corelationbetween deviations of the braking index from n = 3 and deformation for a 1.0 M ⊙ pulsar rotating at frequencies below 160 Hz (notice the expanded y-scale). It is seenthat the shape deformation, even for this most deformable star (i.e. low mass, softEoS), is small at these frequencies and quite unable to reproduce the observed rangeof braking indices. It is important to model pulsar braking indices for frequencies consistent with theaccepted observations. This can be approached in a few different ways. The firstthing to look at is the dependence on C in Eq. 5. For example, in the MDR modelwe have, C ∝ µ sin α , (6)4here µ is the magnetic moment and sin α is the angle of inclination between themagnetic moment and the axis of rotation. It is clear from Eq. 5 that one or bothof these values would have to increase as the pulsar spins down to move toward theobserved braking index values that are less than n = 3. It has been suggested thatthere is some observatinal evidence that sin α is increasing with time [6].Effects on the magnetic field may be a factor for all three accepted torque mech-anisms. The magnetic field may increase, decrease, or change alignment [20]. Thetorque may vary with magnetic field in a way that is not a pure dipole. Plasma out-flow may cause currents in the magnetosphere. Interaction with the magnetospherein general may play in the torque mechanism [13]. Rotationally driven effects suchas a phase transition, density profile, and superfluidity may also affect the magneticfield. The evolution of the magnetic field is most readily applied to the MDR model,but it is also important for the relativistic wind.Particles near the surface of the star can be accelerated to relativistic energies,and thus carry away rotational energy from the pulsar. The wind mechanism has adependence on the magnetic field, and thus all of the considerations explained abovecan affect the braking index due to the wind. The braking index value for the windis n = 1 which is near the lowest measured braking indices shown in Table 1. Sinceall values in Table 1 fall between 1, and 3, it is prudent to consider the wind as apossible torque mechanism affecting braking. The considerations outlined above require knowledge of the origin and distributionof the magnetic field which is not readily available. Also the nature of rotationallydriven effects is not well known. This poses a problem in that there is some speculationinvolved in trying to relate braking index values to changes in the magnetic field orother similar physics in the torque mecahnisms which leads to a phenomenology thatis not necessarily physical.Another approach to this problem considered by Alvarez et. al. [21] is to ex-pand the braking law itself into a polynomial which consists of all assumed torquemechanisms. In this way, we can assume there are no changes in the magnetic field,composition of the star, magnetosphere, etc., and that the pulsar is rotating at lowfrequency. Given that the accurately measured braking indices range in value fromabout 1 - 2.8, it follows that the braking comes form a combination of torque mech-anisms.As shown in [21] the braking law can be expanded as,˙Ω = − s ( t )Ω − r ( t )Ω − g ( t )Ω , (7)where s ( t ), r ( t ), and g ( t ) are functions representing the wind, MDR, and quadrupoletorque mechanisms respectively, and Ω is the rotational frequency.5he polynomial can be used to fit the known braking index values. The brakingindex range of roughly 1 − n = 1) andMDR ( n = 3) should be important. The quadrupole ( n = 5) may not play a roleat low frequencies. Furthermore, the combination of s ( t ) and r ( t ) may constrain themagnetic field. The polynomial solution may also be expanded into high frequencieswhich may give insight into other mechanisms at work such as mass, composition,magnetic field, etc. The problem of braking index can be approached from two ends. We can considervery fast rotation where moment of inertia may dominate the spin evolution, or wecan consider very slow rotation where the physics of the braking mechanism must beadjusted to fit the data. The braking index is dominated by the torque mechanisms atfrequencies below about 200 Hz . Above this frequency, changes in moment of inertiadue to deformation of the star become increasingly important. Unknown compositionand magnetic field dynamics play a crucial role in understanding the braking index,and are important to magnetic dipole radiation and relativistic wind at low frequency.These effects are difficult to model as they are not yet well understood.In order to model braking indices without knowledge of magnetic field evolution orrotationally driven changes in composition, we can construct a polynomial which willfit that data using functions of the known braking index mechanisms at low frequency.This model can be extended to high frequencies, and to include considerations of theunknown values described above. This can also be extended to include physicalequations of state, and mass dependence.A single, cohesive investigation of all the above possibilities is necessary to our un-derstanding of pulsar spin evolution. The resulting parameters may help to constrainthe poorly known micro-physics at work in the cores of rotating neutron stars. Acknowledgement
We express our thanks to the organizers of the CSQCD IV conference for provid-ing an excellent atmosphere which was the basis for inspiring discussions with allparticipants.